It’s a delight to discover that playwright Rachel De-lahay speaks in the same sparky, urban-inflected way as her characters. “Amazing” is a word that recurs frequently in her conversation and she snaps her fingers to underscore her arguments. There’s a verve about her and no wonder: the 29-year-old is the playwright of the moment at the Royal Court, with two pieces opening there this month. It’s little surprise, then, that in a recent interview, the theatre’s artistic director, Vicky Featherstone, named De-lahay as one of the writers she was most excited about.

Peckham: The Soap Opera, on which De-lahay collaborated with Bola Agbaje, finished its run yesterday, so it is Routes, her second full-length piece, that I’ve come to talk about. It covers the same fruitful ground as her award-winning 2011 debut The Westbridge:the melting – and sometimes boiling-over – pot of modern multicultural London. This time, the slant is on immigration, leading to a tightly constructed six-hander that looks at who is and who isn’t allowed to come into and then stay in this country, and the extraordinary lengths to which some would-be immigrants will go.

I wonder if Birmingham-born De-lahay, whose father is from St Kitt’s and whose British-born mother grew up in Pakistan and was married again, to a Jamaican, wrote about the personal experiences of anyone she knew. “Me and my friend did a road trip to Paris – Never drive! It was an absolute rip-off! – and we did the ferry, and I just clocked the guys that hang where the lorries wait to go on. I thought people really will risk everything to get here. The flip-side of that is, if I wasn’t born here, would I do what they’re doing? Is it worth it?’

De-lahay doesn’t consider herself a political playwright (“that’s a really bad thing to say, isn’t it?”), but she is expansive on the Hydra-like topic of current immigration policy. Has she ever been made to feel less than British? “Me and my friends always talk about this. Every non-white person I know admits the world seems easier if you’re white, but no one would pick a different skin colour. Which is amazing…”.

Referring to recent, controversial spot checks by immigration officials, she warms to her theme. “How the hell are they stopping people at stations to say, ‘We think you don’t belong in this country’? I do think it’s a shame that [the immigration debate] is based purely on looks. If officials are hanging outside Stratford station, Anka’s children [Anka is a naturalised British Polish immigrant in Routes] aren’t going to get stopped over mine. They will be white with a British accent, whereas my children will be brown with a British accent. Straightaway you can get, ‘You’re meant to be here, and you’re not.’”

The race debate can come in many forms. I mention a recent storm in a Twitter teacup during which a black playwright accused De-lahay of revisiting “overdone” themes in black playwriting, whatever that might be. She sighs. “I think there’s a fear, which possibly comes from a real place, that’s there’s only room for one [non-white playwright], and if Rachel De-lahay has taken that spot, then that spot’s taken. Laura Wade isn’t going to be kicking off about me having a play on at the Royal Court, Polly Stenham isn’t, because they know there’s room for them.”

Her long route to Routes is as engaging as any of her anecdotes and, winningly, hinges upon her mother, a nurse, going early to bed at “stupid o’clock”. De-lahay trained as an actress, and while doing a play at Coventry and staying at home she found herself bored one evening. “I had a look on the Royal Court website and they had an invitation for Unheard Voices writing groups.” She applied, was accepted into the group for young Muslim writers and the rest is history – plus many drafts of The Westbridge.

She has recently been named a Screen International Star of Tomorrow for her debut film script, currently in development with Film4, about the gripping-sounding true story of a “passion crime” committed by a girl with whom she went to school. Television, however, excites her more than film, and she has a comedy-drama television project in the pipeline. “I want as many people to hear the stories as possible. I haven’t been to the cinema in ages, and I feel like I’m probably not alone in that. I don’t know who you’re telling those stories to if it’s in a cinema. Who are the people that are still going?”

It’s all very well being one of the most buzzed-about young playwrights, but does that translate into being able to pay her bills? “Just about,” she says wryly, adding that she worked full-time as a supervisor in a clothes shop until last year. “My flatmate’s a stand-up comedian, and month by month the joke is we don’t have a standing order for any of our bills. It can’t just come out, because what if it’s not there? That’s the luxury: if you can pay your bills with a standing order, you’re living the dream! That’s amazing!” I confidently predict that De-lahay will be able to set up those standing orders before the year is out.